Hermite-Hadamard inequality for new generalized conformable fractional operators

Tahir Ullah Khan; Muhammad Adil Khan; Tahir Ullah Khan; Muhammad Adil Khan

doi:10.3934/math.2021002

AIMS Mathematics

2021, Volume 6, Issue 1: 23-38. doi: 10.3934/math.2021002

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Research article

Hermite-Hadamard inequality for new generalized conformable fractional operators

Tahir Ullah Khan ¹,
Muhammad Adil Khan ^{2
,
,}

1.
Department of Mathematics, University of Peshawar, Peshawar 25000, Pakistan
2.
Department of Mathematics, Huzhou University, Huzhou 313000, China

Received: 11 August 2020 Accepted: 17 September 2020 Published: 28 September 2020
MSC : 26A51, 26D15, 26D20

This paper is concerned to establish an advanced form of the well-known Hermite-Hadamard (HH) inequality for recently-defined Generalized Conformable (GC) fractional operators. This form of the HH inequality combines various versions (new and old) of this inequality, containing operators of the types Katugampula, Hadamard, Riemann-Liouville, conformable and Riemann, into a single form. Moreover, a novel identity containing the new GC fractional integral operators is proved. By using this identity, a bound for the absolute of the difference between the two rightmost terms in the newly-established Hermite-Hadamard inequality is obtained. Also, some relations of our results with the already existing results are presented. Conclusion and future works are presented in the last section.
- generalized conformable fractional operators,
- Riemann-Liouville operators,
- conformable integral,
- Hermite-Hadamard inequality
Citation: Tahir Ullah Khan, Muhammad Adil Khan. Hermite-Hadamard inequality for new generalized conformable fractional operators[J]. AIMS Mathematics, 2021, 6(1): 23-38. doi: 10.3934/math.2021002

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Abstract

This paper is concerned to establish an advanced form of the well-known Hermite-Hadamard (HH) inequality for recently-defined Generalized Conformable (GC) fractional operators. This form of the HH inequality combines various versions (new and old) of this inequality, containing operators of the types Katugampula, Hadamard, Riemann-Liouville, conformable and Riemann, into a single form. Moreover, a novel identity containing the new GC fractional integral operators is proved. By using this identity, a bound for the absolute of the difference between the two rightmost terms in the newly-established Hermite-Hadamard inequality is obtained. Also, some relations of our results with the already existing results are presented. Conclusion and future works are presented in the last section.

References

[1]	I. Podlubny, Fractional differential equations: An introduction to fractional derivatives, fractional differential equations, to methods of their solution and some of their applications, Academic Press, 1999.
[2]	E. C. De Oliveira, J. A. T. Machado, Review of definitions for fractional derivatives and integrals, Math. Probl. Eng., 2014 (2014), 1-6.
[3]	A. Atangana, On the new fractional derivative and application to nonlinear Fisher's reaction-diffusion equation, Appl. Math. Comput., 273 (2016), 948-956.
[4]	A. Atangana, I. Koca, On the new fractional derivative and application to nonlinear Baggs and Freedman model, J. Nonlinear Sci. Appl., 9 (2016), 2467-2480. doi: 10.22436/jnsa.009.05.46
[5]	T. U. Khan, M. A. Khan, Generalized conformable fractional operators, J. Comput. Appl. Math., 346 (2019), 37-389.
[6]	T. Abdeljawad, On conformable fractional calculus, J. Comput. Appl. Math., 279 (2015), 57-66. doi: 10.1016/j.cam.2014.10.016
[7]	R. Khalil, M. Al Horani, A. Yousef, M. Sababheh, A new definition of fractional derivative, J. Comput. Appl. Math., 264 (2014), 65-70. doi: 10.1016/j.cam.2014.01.002
[8]	A. A. Kilbas, M. H. Srivastava, J. J. Trujillo, Theory and application of fractional differential equations, North-Holland Mathematics Studies, 2006.
[9]	K. S. Miller, B. Ross, An Introduction to the fractional calculus and fractional differential equations, Wiley, 1993.
[10]	A. Atangana, B. S. T. Alkahtani, Analysis of the Keller-Segel model with a fractional derivative without singular kernel, Entropy, 17 (2015), 4439-4453. doi: 10.3390/e17064439
[11]	A. Atangana, I. Koca, Chaos in a simple nonlinear system with Atangana-Baleanu derivatives with fractional order, Chaos Solitons & Fractals, 89 (2016), 447-454.
[12]	A. Atangana, D. Baleanu, New fractional derivatives with non-local and non-singular kernel: Theory and application to heat transfer model, Therm. Sci., 20 (2016), 763- 769. doi: 10.2298/TSCI160111018A
[13]	U. N. Katugampola, New approach to a generalized fractional integral, Appl. Math. Comput., 218 (2011), 860-865.
[14]	P. O. Mohammed, On new trapezoid type inequalities for h-convex functions via generalized fractional integral, Turk. J. Anal. Number Theory, 6 (2018), 125-128. doi: 10.12691/tjant-6-4-5
[15]	S. S. Dragomir, C. Pearce, Selected topics on Hermite-Hadamard inequalities and applications, RGMIA Monographs, Victoria University, 2000.
[16]	T. Abdeljawad, P. O. Mohammed, A. Kashuri, New modified conformable fractional integral inequalities of Hermite-Hadamard type with applications, J. Funct. Space., 2020 (2020), 1-14.
[17]	P. O. Mohammed, I. Brevik, A new version of the Hermite-Hadamard inequality for Riemann-Liouville fractional integrals, Symmetry, 12 (2020), 1-11.
[18]	M. A. Khan, N. Mohammad, E. R. Nwaeze, et al. Quantum Hermite-Hadamard inequality by means of a Green function, Adv. Differ. Equ., 2020 (2020), 1-20. doi: 10.1186/s13662-019-2438-0
[19]	P. O. Mohammed, T. Abdeljawad, Integral inequalities for a fractional operator of a function with respect to another function with nonsingular kernel, Adv. Differ. Equ., 2020 (2020), 1-19. doi: 10.1186/s13662-019-2438-0
[20]	M. A. Khan, Y. Khurshid, S. S. Dragomir, R. Ullah, Inequalities of the HermiteHadamard type with applications, Punjab Univ. J. Math., 50 (2018), 1-12.
[21]	M. A. Khan, Y. M. Chu, T. U. Khan, J. Khan, Some new inequalities of HermiteHadamard type for s-convex functions with applications, Open Math., 15 (2017), 1414-1430. doi: 10.1515/math-2017-0121
[22]	M. A. Khan, Y. M. Chu, A. Kashuri, R. Liko, G. Ali, New Hermite-Hadamard inequalities for conformable fractional integrals, J. Funct. Space., 2018 (2018), 1- 9.
[23]	A. Iqbal, M. A. Khan, S. Ullah, Y. M. Chu, A. Kashuri, Hermite-Hadamard type inequalities pertaining conformable fractional integrals and their applications, AIP adv., 8 (2018), 1-18.
[24]	M. A. Khan, Y. Khurshid, T. S. Du, Y. M. Chu, Generalization of Hermite-Hadamard Type Inequalities via Conformable Fractional Integrals, J. Funct. Space., 2018 (2018), 1-12.
[25]	Y. Khurshid, M. A. Khan, Y. M. Chu, Conformable fractional integral inequalities for GG- and GA-convex functions, AIMS Mathematics, 5 (2020), 5012-5030. doi: 10.3934/math.2020322
[26]	P. O. Mohammed, M. Z. Sarikaya, On generalized fractional integral inequalities for twice differentiable convex functions, J. Comput. Appl. Math., 372 (2020), 1-15.
[27]	M. A. Khan, T. U. Khan, Y. M. Chu, Generalized Hermite-Hadamard type inequalities for quasi-convex functions with applications, Journal of Inequalities & Special Functions, 11 (2020), 24-42.
[28]	M. A. Khan, A. Iqbal, M. Suleman, Y. M. Chu, Hermite-Hadamard type inequalities for fractional integrals via Green's function, J. Inequal. Appl., 2018 (2018), 1-15. doi: 10.1186/s13660-017-1594-6
[29]	M. A. Khan, Y. Khurshid, Y. M. Chu, Hermite-Hadamard type inequalities via the Montgomery identity, Communications in Mathematics and Applications, 10 (2019), 85-97.
[30]	A. Iqbal, M. A. Khan, N. Mohammad, E. R. Nwaeze, Y. M. Chu, Revisiting the Hermite-Hadamard fractional integral inequality via a Green function, AIMS Mathematics, 5 (2020), 6087-6107. doi: 10.3934/math.2020391
[31]	Y. M. Chu, M. A. Khan, T. Ali, S. S. Dragomir, Inequalities for α-fractional differentiable functions, J. Inequal. Appl., 2017 (2017), 1-12. doi: 10.1186/s13660-016-1272-0
[32]	J. Han, P. O. Mohammed, H. Zeng, Generalized fractional integral inequalities of Hermite-Hadamard-type for a convex function, Open Math., 18 (2020), 794-806. doi: 10.1515/math-2020-0038
[33]	M. Z. Sarikaya, E. Set, H. Yaldiz, N. Ba?ak, Hermite-Hadamard's inequalities for fractional integrals and related fractional inequalities, Math. Comput. Model., 57 (2013), 2403-2407.
[34]	Y. M. Chu, M. A. Khan, T. U. Khan, T. Ali, Generalizations of Hermite-Hadamard type inequalities for MT-convex functions, J. Nonlinear Sci. Appl., 9 (2016), 4305- 4316. doi: 10.22436/jnsa.009.06.72
[35]	Y. Khurshid, M. A. Khan, Y. M. Chu, Hermite-Hadamard-Fejer inequalities for conformable fractional integrals via preinvex functions, J. Funct. Space., 2019 (2019), 1-9.
[36]	P. O. Mohammed, M. Z. Sarikaya, D. Baleanu, On the generalized Hermite-Hadamard inequalities via the tempered fractional integrals, Symmetry, 12 (2020), 1-17.
[37]	M. A. Khan, T. Ali, S. S. Dragomir, M. Z. Sarikaya, Hermite-Hadamard type inequalities for conformable fractional integrals, RACSAM Rev. R. Acad. A, 112 (2018), 1033-1048.
[38]	M. Jleli, D. O'regan, B. Samet, On Hermite-Hadamard type inequalities via generalized fractional integrals, Turk. J. Math., 40 (2016), 1221-1230. doi: 10.3906/mat-1507-79
[39]	M. A. Khan, S. Begum, Y. Khurshid, Y. M. Chu, Ostrowski type inequalities involving conformable fractional integrals, J. Inequal. Appl., 2018 (2018), 1-14. doi: 10.1186/s13660-017-1594-6
[40]	E. R. Nwaeze, M. A. Khan, Y. M. Chu, Fractional inclusions of the HermiteHadamard type of m-polynomial convex interval-valued functions, Adv. Differ. Equ., 2020 (2020), 1-17. doi: 10.1186/s13662-019-2438-0
[41]	A. Guessab, G. Schmeisser, Convexity results and sharp error estimates in approximate multivariate integration, Math. Comput., 73 (2004), 1365-1384.
[42]	A. Guessab, G. Schmeisser, Sharp integral inequalities of the Hermite-Hadamard type, J. Approx. Theory, 115 (2002), 260-288. doi: 10.1006/jath.2001.3658

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